Cooperative Dynamics of Oscillator Community

A phase description of systems of many limit cycle oscillators is established. As a result, the systems are reduced to populations of simple elements called rings, their mutual coupling depending only on the phase difference between the interacting pair. A few typical lattice models of rings are proposed. Besides pattern formation and turbulencelike behavior, our ring models, as statistical cooperative systems, can exhibit curious phase transitions with the order parameter essentially of dynamic origin. Finally, such fields of rings are generalized in order to simulate excitable media, and also to incorporate equilibrium cooperative fields of some known universality classes.